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Simplifying -2z2 = -5z + -4 Reorder the terms: -2z2 = -4 + -5z Solving -2z2 = -4 + -5z Solving for variable 'z'. Reorder the terms: 4 + 5z + -2z2 = -4 + -5z + 4 + 5z Reorder the terms: 4 + 5z + -2z2 = -4 + 4 + -5z + 5z Combine like terms: -4 + 4 = 0 4 + 5z + -2z2 = 0 + -5z + 5z 4 + 5z + -2z2 = -5z + 5z Combine like terms: -5z + 5z = 0 4 + 5z + -2z2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2 + -2.5z + z2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -2.5z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + -2.5z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -2.5z + z2 = 0 + 2 -2.5z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 -2.5z + z2 = 2 The z term is -2.5z. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5z + 1.5625 + z2 = 2 + 1.5625 Reorder the terms: 1.5625 + -2.5z + z2 = 2 + 1.5625 Combine like terms: 2 + 1.5625 = 3.5625 1.5625 + -2.5z + z2 = 3.5625 Factor a perfect square on the left side: (z + -1.25)(z + -1.25) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (z + -1.25) equal to 1.887458609 and -1.887458609.Subproblem 1
z + -1.25 = 1.887458609 Simplifying z + -1.25 = 1.887458609 Reorder the terms: -1.25 + z = 1.887458609 Solving -1.25 + z = 1.887458609 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + z = 1.887458609 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + z = 1.887458609 + 1.25 z = 1.887458609 + 1.25 Combine like terms: 1.887458609 + 1.25 = 3.137458609 z = 3.137458609 Simplifying z = 3.137458609Subproblem 2
z + -1.25 = -1.887458609 Simplifying z + -1.25 = -1.887458609 Reorder the terms: -1.25 + z = -1.887458609 Solving -1.25 + z = -1.887458609 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + z = -1.887458609 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + z = -1.887458609 + 1.25 z = -1.887458609 + 1.25 Combine like terms: -1.887458609 + 1.25 = -0.637458609 z = -0.637458609 Simplifying z = -0.637458609Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.137458609, -0.637458609}
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